SOLUTION: How many different "words" can be constructed using the word STATISTICS, if the "word" should end with a vowel only?

There are two ways to pick the vowel for the "word" to end in, A or I.

Case 1. We pick the A for the "word" to end in.

Then the letters to go before the A on the end are the distinguishable
permutations of STTISTICS.  

There are 3 indistinguishable S's,
          3 indistinguishable T's, and 
          2 indistinguishable I's

That's 9!/(3!3!2!) = 5040 ways.

Case 2. We pick an I for the "word" to end in.

Then the letters to go before the A on the end are the distinguishable
permutations of STATSTICS.  

There are 3 indistinguishable S's, and
          3 indistinguishable T's 
          
That's 9!/(3!3!) = 10080 ways.

Grand total: 5040+10080 = 15120 ways.

Edwin